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Irrotationality somewhere = irrotationality everywhere?

  1. Dec 20, 2013 #1
    I'm learning about 2D inviscid irrotational flows of constant density. In the example of flow past a cylinder there is the sentence "since the flow is irrotational as r tends to infinity, it is irrotational everywhere" and I can't get my hear around that.

    Why is this the case?

    Irrotational means that vorticity is zero, and in the case of an inviscid flow of constant density Kelvin's circulation theorem means that it remains zero as you follow any loop of fluid that initially has zero circulation. However I don't see why it is guaranteed that the flow is irrotational near the cylinder if it is irrotational at infinity.

    Thanks for any help!
  2. jcsd
  3. Dec 20, 2013 #2


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    Basically, because flow is inviscid, vorticity is zero within the flow. But you can still have circulation. Imagine that the cylinder rotates. In that case, the flow around it will have circulation. However, because vorticity is zero in the fluid, that circulation is exactly the same around any closed contour containing cylinder in its interior. That's how you get Magnus Effect.

    What they are telling you in the text is that for your problem, circulation being zero at infinity implies circulation is zero everywhere throughout the fluid. Therefore, fluid is irrotational.
  4. Dec 21, 2013 #3


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    Consider Helmholtz's theorem as it relates to this situation. You can derive it by taking the curl of the Euler equation and it will show that for a fluid element moving with the flow, tyke vorticity is constant as it convects with the flow for an ideal fluid under the action of conservative body forces.

    So, knowing that, if the flow is irrotational far upstream of your cylinder (and that would be true of a uniform free stream), any fluid that starts upstream (and therefore all fluid) will remain at zero vorticity, even under the influence of said cylinder.
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